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Excess 2^n-1 encoding

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• 2's complement encoding
  • The 2's complement encoding makes the assumption that:

    The representation for the value 0 is: 00000000

    This results in the following assignment scheme of binary numbers and values:

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• Excess 2ⁿ⁻¹ encoding
  • There is another way to obtain a representation of signed numbers

  • The Excess 2ⁿ⁻¹ encoding makes the assumption that:

    The representation for the value 0 is: 1000...0000 The binary number 1000...0000 of n bits is the representation for the value 2n−1 (Hence the encoding is called Excess 2n−1 encoding)

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• Example: Excess 2ⁿ⁻¹ encoding with n = 8 bits
  • The Excess 128 (= 2⁸⁻¹) encoding scheme makes the assumption that:

    The representation for the value 0 is: 10000000

    This results in the following assignment scheme of binary numbers and values:

  • The Excess 128 encoding scheme is as follows:

            Code      Value
            ================
            00000000  -128  <--- smallest negative value with 8 bits (-27)
            00000001  -127
            .....
            01111000    -8
            01111001    -7
            01111010    -6
            01111011    -5
            01111100    -4
            01111101    -3
            01111110    -2
            01111111    -1
            10000000     0
            10000001     1
            10000010     2
            10000011     3
            10000100     4
            10000101     5
            10000110     6
            10000110     6
            10000111     7
            10001000     8
            .....
            11111111   127  <--- largest positive value with 8 bits (27-1)
    

    Note:

    The representation 00000000 represents the value −128 !!! This is not a typo !!!

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  • Wikipedia calls this scheme offset binary: click here

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• Another possible offset: 2ⁿ⁻¹ − 1
  • There is another commonly used choice for the representation of the value 0

  • The Excess 2ⁿ⁻¹ −1 encoding makes the assumption that:

    The representation for the value 0 is: 011111...11111 The binary number 011111...11111 of n bits is the representation for the value 2n−1 −1 (Hence the encoding is called Excess 2n−1 −1 encoding)

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• Example: Excess 2ⁿ⁻¹−1 encoding with n = 8 bits
  • The Excess 127 (= 2⁸⁻¹−1) encoding scheme makes the assumption that:

    The representation for the value 0 is: 01111111

    This results in the following assignment scheme of binary numbers and values:

  • The Excess 127 encoding scheme is as follows:

            Code      Value
            ================
            00000000  -127  <--- smallest negative value with 8 bits (-27)
            00000001  -126
            .....
            01111000    -7
            01111001    -6
            01111010    -5
            01111011    -4
            01111100    -3
            01111101    -2
            01111110    -1
            01111111     0
            10000000     1
            10000001     2
            10000010     3
            10000011     4
            10000100     5
            10000101     6
            10000110     7
            10000110     8
            10000111     9
            10001000    10
            .....
            11111111   128  <--- largest positive value with 8 bits (27-1)
    

    Note:

    The representation 00000000 represents the value −127 !!! This is not a typo !!!

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  • Note:

    We will see the excess 127 encode again when we discuss floating point number representation Excess 127 is used to represent the exponent of the single precision IEEE 754 standard.